Let \(\rho\) be a root of: \(x^2-2x-1=0\)

Define a function \(F(n) \mid n \in \mathbb N\) by the recurrence relation:

\(F(0)=0\)

\(F(1)=1\)

\(F(n)= 2F(n-1)+F(n-2) \mid n \geq 2\)

Prove that for all integers \(n\geq1\):

\[\rho^n = F(n-1)+F(n)\rho\]

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